Given F(x,y) = , find a function f such that gradient f = F and use it to evaluate integral over C of F * dr where C is the arc of the parabola y = 2x^2 from (-1,2) to (2,8)

Question

Given F(x,y) = <x^2, y^2>, find a function f such that gradient f = F and use it to evaluate integral over C of F * dr where C is the arc of the parabola y = 2x^2 from (-1,2) to (2,8)

anonymous · Answer

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anonymous · Answer

I have been able to find f. $$f = (1/3)x^3 +  (1/3)y^3$$

anonymous · Answer

what I am not to sure in how to apply is the Fundamental Theorem of Line Integrals as the theorem is based on vectors, vector r, and I am given a line this time, y = 2x^2.

fwizbang · Answer

Since F is the gradient of  f,  the path you take between the two points doesn't matter, so you can calculate the line integral by evaluating f at the two end points and subtracting.